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Physical Zero-Knowledge Proof Protocol for Topswops

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Information Security Practice and Experience (ISPEC 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13620))

Abstract

Suppose that a sequence of n cards, numbered 1 to n, is placed face up in random order. Let k be the number on the first card in the sequence. Then take the first k cards from the sequence, rearrange that subsequence of k cards in reverse order, and return them to the original sequence. Repeat this prefix reversal until the number on the first card in the sequence becomes 1. This is a one-player card game called Topswops. The computational complexity of Topswops has not been thoroughly investigated. For example, letting f(n) denote the maximum number of prefix reversals for Topswops with n cards, values of f(n) for \(n\ge 20\) remain unknown. In general, there is no known efficient algorithm for finding an initial sequence of n cards that requires exactly \(\ell \) prefix reversals for any integers n and \(\ell \). In this paper, we propose a physical zero-knowledge proof protocol that allows a prover to convince a verifier that the prover knows an initial sequence of n cards that requires \(\ell \) prefix reversals without leaking knowledge of that sequence.

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Notes

  1. 1.

    https://oeis.org/A000375.

  2. 2.

    https://oeis.org/A123398.

  3. 3.

    According to Zimmermann’s Programming Contests (http://azspcs.com/Contest/Cards/FinalReport, accessed 16 Aug 2022), four initial sequences which require 221 steps for \(n=19\) were discovered in 2011. Hence, it seems that at that time the lower bounds on f(19) and g(19) were known to be 221 and 4, respectively.

  4. 4.

    At the present moment, of course, it is unknown whether such an initial sequence exists, i.e., whether \(f(20) \ge 250\) or \(f(20)=249\).

  5. 5.

    More precisely, it is specified as a card-based protocol formulated in the standard model of card-based cryptography [36, 37]. (Note that there are other models, e.g., [26, 27, 39, 40].).

  6. 6.

    The upper and lower bounds were known to be \(\frac{17}{16}n \le h(n) \le \frac{5n+5}{3}\) [7] in 1979.

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Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (JP18H05289, JP21K11881).

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Authors and Affiliations

  1. Toshiba Corporation, 1 Komukai-Toshiba-cho, Saiwai-ku, Kawasaki, Japan

    Yuichi Komano

  2. Tohoku University, 6-3 Aramaki-Aza-Aoba, Aoba-ku, Sendai, Japan

    Takaaki Mizuki

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  1. Yuichi Komano
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  1. University of Aizu, Fukushima, Japan

    Chunhua Su

  2. Athens University of Economics and Business, Athens, Greece

    Dimitris Gritzalis

  3. Università degli Studi di Milano, Milan, Italy

    Vincenzo Piuri

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Komano, Y., Mizuki, T. (2022). Physical Zero-Knowledge Proof Protocol for Topswops. In: Su, C., Gritzalis, D., Piuri, V. (eds) Information Security Practice and Experience. ISPEC 2022. Lecture Notes in Computer Science, vol 13620. Springer, Cham. https://doi.org/10.1007/978-3-031-21280-2_30

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